Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces
 Claudio Vidal^{1}Email author
DOI: 10.1155/2009/380568
© Claudio Vidal 2009
Received: 20 November 2008
Accepted: 10 June 2009
Published: 16 June 2009
Abstract
The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.
1. Introduction
assuming that this system possesses a bounded solution with some property of stability. In (1.1) , and denotes an abstract phase space which we will define later.
The abstract space was introduced by Hale and Kato [1] to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham [2], Hino et al. [3]. The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami [4, 5], Elaydi et al. [6], Cuevas and Pinto [7, 8], Cuevas and Vidal [9], and Cuevas and Del Campo [10].
As usual, we denote by , , and the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let be the dimensional complex Euclidean space with norm . the set .
If is a function, we define for , the function by , . Furthermore is the function given for , with .
 (A)There is a positive constant and nonnegative functions and on with the property that is a function, such that , then for all , the following conditions hold:
 (i)
,
 (ii)
,
 (iii)
.
 (i)
 (B)
The space is a Banach space.
We need the following property on .
 (C)
The inclusion map is continuous, that is, there is a constant , such that , for all , where represents the bounded functions from into .
Axiom (C) says that any element of the Banach space of the bounded functions equipped with the supremum norm is on .
Remark 1.1.
 (C')
If a uniformly bounded sequence in converges to a function compactly on (i.e., converges on any compact discrete interval in ) in the compactopen topology, then belong to and as .
Remark 1.2.
We will denote by ( , and ) or simply by , the solution of (1.1) passing through , that is, , and the functional equation (1.1) is satisfied.
During this paper we will assume that the sequences and are bounded. The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. In Section 4 we analyze separately the cases where is periodic and when it is almost periodic. Thus, in Section 4.1 assuming that the system (1.1) is periodic and the existence of a bounded solution (particular solution) which is uniformly stable and the phase space satisfies only the axioms (A)–(C), we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, in Section 4.2 considering that system (1.1) is almost periodic and the existence of a bounded solution and whenever the phase space satisfies the axioms (A)–(C), but here it is also necessary that verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system (1.1) has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.
In [11, 12] the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in [13] the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya's because, firstly, in our work we consider both cases, namely, when is periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of (1.1) with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.
Furthermore, we cite the articles [14–16] which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.
2. Fading Memory Spaces and Implications
for .
Definition 2.1.
A phase space that satisfies axioms (A)(B) and ( ) or ( ) and such that the semigroup is strongly stable is called a fading memory space.
Remark 2.2.
Remember that a strongly continuous semigroup is strongly stable if for all , as .
Thus, we have the following result.
Lemma 2.3.
Let , with , where is a fading memory space. If as , then as .
Proof.
Then, by definition as because . On the other hand, by hypothesis, as , so it follows from Axiom (C') that . Therefore, we conclude that as .
3. Notations and Preliminary Results
In this section, we review the definitions of (uniformly) almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12].
Definition 3.1.
is called the translation number of . We will denote by the set of all such sequences. We will write that is a.p. if .
Definition 3.2.
where is an almost periodic sequence, and as . We will denote by the set of all such sequences. We will write that is a.a.p. if .
In general, we will consider a Banach space.
Definition 3.3.
Denote by all such sequences, and is said to be an almost periodic (a.p.) in .
Definition 3.4.
A sequence , (or ), , equivalently, a function (or, ) is called asymptotically almost periodic if , where and (or, ) satisfying as (or, ). Denote by (or all such sequences, and is said to be an asymptotically almost periodic on (or on ) (a.a.p.) in .
Remark 3.5.
Almost periodic sequences can be also defined for any sequence ( ) or by requiring that consecutive integers are in .
Definition 3.6.
is called the translation number of . We will denote by the set of all such sequences. In brief we will write that is u.a.p. if .
Definition 3.7.
for some sequence , where is any compact set in .
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12]. With the objective to make this manuscript self contained we decided to include the majority of the proofs.
 (a)
If is an a.p. sequence, then there exists an almost periodic function such that for .
 (b)
If is an a.p. function, then is an a.p. sequence.
Lemma 3.9. (a) If is an a.p. sequence, then is bounded.
(b) is an a.p. sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limits sequence is also an almost periodic sequence.
where for .
(d) , (or, ) is an a.a.p. sequence if and only if for any sequence (or, ) such that and as (or, as ), there exists a subsequence such that converges uniformly on (or ) as .
Lemma 3.10.
where is an a.p. sequence while as , is unique.
Lemma 3.11.
Let be almost periodic in uniformly for and continuous in . Then is bounded and uniformly continuous on for any compact set in .
Lemma 3.12.
Let be the same as in the previous lemma. Then, for any sequence , there exist a subsequence of and a function continuous in such that uniformly on as , where is any compact set in . Moreover, is also almost periodic in uniformly for .
Lemma 3.13.
Let be the same as in the previous lemma. Then, there exists a sequence , as such that uniformly on as , where is any compact set in .
Lemma 3.14.
Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in . Then is almost periodic in .
Lemma 3.15.
Let be almost periodic in uniformly for and continuous in , and let be an almost periodic sequence in such that for all , where is a compact set in and for . Then is almost periodic in .
Remark 3.16.
If is a.a.p., then the decomposition , in the definition of an a.a.p. function, is unique (see [18]).
4. Existence of Almost Periodic Solutions
From now on we will assume that the system (1.1) has a unique solution for a given initial condition on and without loss of generality , thus .
 (H1)
is continuous in the second variable for any fixed .
 (H2)
System (1.1) has a bounded solution , passing through , , that is, .
For this bounded solution , there is an such that for all . So, we will have to assume that for all , and . Next, we will point out the definitions of stability for functional difference equations adapting it from the continuous case according to Hino et al. in [3].
Definition 4.1.
 (i)
stable, if for any and any integer , there is such that implies that for all , where is any solution of (1.1);
 (ii)
uniformly stable, abbreviated as " '', if for any and any integer , there is ( does not depend on ) such that implies that for all , where is any solution of (1.1);
 (iii)
uniformly asymptotically stable, abbreviated as " '', if it is uniformly stable and there is such that for any , there is a positive integer such that if and , then for all , where is any solution of (1.1);
 (iv)
globally uniformly asymptotically stable, abbreviated as " '', if it is uniformly stable and as , whenever is any solution of (1.1).
Remark 4.2.
 (iii)
is , if it is uniformly stable, and there exists such that if and , then as , where is any solution of (1.1).
4.1. The Periodic Case
 (H3)
The function in (1.1) is periodic in , that is, there exists a positive integer such that for all .

( ) The sequences and in Axiom (A)(iii) are bounded by and , respectively and .
Lemma 4.3.
Suppose that condition ( ) holds. If is a bounded solution of (1.1) such that , then is also bounded in .
Proof.
Lemma 4.4.
Suppose that condition ( ) holds. Let be a sequence in such that for all . Assume that as for every and , then in as for each . In particular, if as uniformly in , then in as uniformly in .
Proof.
for each . Therefore, we have concluded the proof.
Theorem 4.5.
Suppose that condition ( ) and (H1)–(H3) hold. If the bounded solution of (1.1) is , then is an a.a.p. sequence in , equivalently, (1.1) has an a.a.p. solution.
Proof.
through . It is clear that if is , then is also with the same pair as the one for .
This implies that for any positive integer sequence , as , there is a subsequence of for which converges uniformly on as . Thus, the conclusion of the theorem follows from Lemma 3.9(d).
Before proving our following result we remark that if is a.a.p. then there are unique sequences such that , with a.p. and as as . By Lemma 3.9(a) it follows that is bounded and thus . Hence, by Axiom (C) we must have that for all . In particular, for all .
Theorem 4.6.
Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then system (1.1) has an a.p. solution, which is also .
Proof.
as , we have for , that is, the system (1.1) has an almost periodic solution, and so we have proved the first statement of the theorem.
for all , which implies that for all if because is arbitrary. This proves that is .
In the case when we have an asymptotically stable solution of (1.1) we obtain the following result.
Theorem 4.7.
Suppose that and (H1)–(H3) hold and the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period for some positive integer , which is also .
Proof.
which implies that for all because is a.p.
For the integer sequence , , we have . Then uniformly for all as , and again by Lemma 4.4, uniformly in as . Since , we have for , which implies that (1.1) has a periodic solution of period .
as if , because , , and satisfy (1.1). This completes the proof.
Finally, if the particular solution is , we will prove that system (1.1) has a periodic solution.
Theorem 4.8.
Suppose that and (H1)–(H3) hold and that the bounded solution of (1.1) is , then the system (1.1) has a periodic solution of period .
Proof.
By Theorem 4.5, is a.a.p. Then ), where ( ) is an a.p. sequence and as . Notice that is also a solution of (1.1) satisfying . Since is , we have that as , which implies that for all . Using same technique as in the proof of Theorem 4.7, we can show that is a periodic solution of (1.1).
4.2. The Almost Periodic Case
 (H4)
the function in (1.1) is almost periodic in uniformly in the second variable.
By we denote the uniform closure of , that is, . Note that by Lemma 3.12 and by Lemma 3.13.
Lemma 4.9.
Suppose that Axiom (C) is true, and that is an a.p. sequence with , then is a.p.
Proof.
Lemma 4.10.
Suppose that is a fading memory space and is a.a.p. with , then is a.a.p.
Proof.
Since is a.a.p. there are unique sequences and such that is a.p. and as . Then by Lemma 4.9 it follows that is a.p., and by Lemma 2.3 it follows that as . Therefore, is a.a.p.
Theorem 4.11.
Suppose that conditions , (H1)(H2), and (H4) hold and that is a fading memory space. If the bounded solution of (1.1) is an a.a.p. sequence, then the system (1.1) has an a.p. solution.
Proof.
and from the previous considerations the first term of the righthand side of (4.23) tends to zero as and since as , we have that for all , which implies that (1.1) has an a.p. solution passing through , where for .
We are now in a position to prove the following result.
Theorem 4.12.
Suppose that the assumptions , (H1), (H2), and (H4) hold, and that is a fading memory space. If the bounded solution of (1.1) is , then is a.a.p. Consequently, (1.1) has an a.p. solution which is .
Proof.
and hence by Axiom A(ii) for all if . This implies that the bounded solution of (1.1) is a.a.p. by Lemma 3.9(d). Furthermore, (1.1) has an a.p. solution, which is by Theorem 4.11. This ends the proof.
Appendix
Authors’ Affiliations
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